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ACR No. L6A22 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS WAlTIME II REPORT ORIGINALLY ISSUED March 1946 as Advance Confidential Report L6A22 APPROXIMATE FORMULAS FOR THE COMPUTATION OF TURBULiNT BOUNDARYLAYER MOMETUM THICKNESSES IN COMPRESSIBLE FLOWS By Neal Tetervin Langley Memorial Aeronautical Laboratory Langley Field, Va. '*, : .i : ; .. . WASHINGTON NACA WARTIME REPORTS are reprints of papers originally issued to provide rapid distribution of advance research results to an authorized group requiring them for the war effort. They were pre viously held under a security status but are now unclassified. Some of these reports were not tech nically edited. All have been reproduced without change in order to expedite general distribution. L 119 DOCUMENTS DEPARTMENT WAt l)(1 Digitized by the Internet Archive in 2011 with funding from University of Florida, George A. Smathers Libraries with support from LYRASIS and the Sloan Foundation http://www.archive.org/details/approximateformu001ang  1 14 i  T.'A .':. Ho. L6A22 :. IC'ITLI ADVISORY CO'.nITTEO FOR AERONAUTICS r '. '_,' .'' ,,F'iZE;'TIm .'r F;I'' 9?, CT NPrROXIr!ATE7 FOR~1T AS FOR THE COMPUT TIOI OPF 7TR E'TIUiT EPOn'TDARYL AYER TMOMBTUM THICv.ICSSES IIH 00 ,PRESS IBE FLOcVS By 'Ieal Tetervin SUMMA L RY Apprczimate formulas for the comut'tion cf the moment'ru thicknesses of turbulent boundary layers on twodimensional bodies, on bodies of revolution at zero angle of attack, and on the inner surfaces of round channels all in compressible flow are given in the form of irtecrls that c0n be conveniently computed. The formulas involve the assumt"itions t.at the mc.mentWun thickness may be coi~puted b u.se of a bcundarylayer velocity profile which is fixed: .and that skinfriction formulas fcr flat plates nay be used in the computation of 'oundarylayer thicknesses in flow with pressure gradients. The effect of density changes on the ratio of the disolacement thickness to the momentinm thickness of the boundary lIver is tken intc account. Use is made of the experimental finding that the skinfriction coefficient for turbulent flow is independent of i'ach number. The comoutstions indicate that the effect of density changes on the momentum thickness in flows with pressure rra.'ients is smRall for subsonic flovs. TI.TRODUCTIO'7 A nLLmber of methods are available for the computation of boundarylayer momentum thicknesses for inccmoressible flow. ThE increasing importance of flows at Mach numbers approaching and exceeding 1 has emphasized the need cf formulas that r.oul:l ?alke possible the comparatively rapid computation of boundarylayer momentum thicknesses for corpressible flows. The purpose of the present work is therefore to provide approximate formulas for the compu tation of boundarylayerthickness parameters for 2 CONFIDENTIAL NACA ACR No. L6A22 compressible flows. The present work furnishes no new information concerning the boundarylayer shape, skin friction coefficient, position of the transition point, or likelihood of boundarylayer separation. Approximate formulas for the computation of the momentum thicknesses of turbulent boundary layers on twodimensional bodies, on bodies of revolution at zero angle of attack, and on the inner surfaces of round channels all in compressible flow are given in the form of integrals that can be conveniently computed. The approximate formulas contain the assumptions that the momentum thickness may be computed by use of a boundary layer velocity profile which is fixed during the inte gration and that skinfriction formulas for flat plates may be used in the computation of boundarylayer momentum thicknesses for flow with pressure gradients. The formulas are applicable to all unseparated, turbulent boundary layers and in special cases to laminar boundary layers. The numerical values of the ratio of the dis placement thickness to the momentum thickness, a ratio that appears in the momentum equation and that is capable of specifying approximately the velocity distribution through the turbulent boundary layer in incompressible flow, are corrected for density changes in the boundary layer by use of lowspeed velocity distributions. Use is made of the.experimental finding that the skinfriction coefficient in turbulent flow is independent of Mach number. The problem of computing the boundarylayer momentum thickness for compressible flow has been treated by Young and Winterbottom (reference 1), who integrated the boundarylayer momentum equation for laminar flow by using the skinfriction relation from the Pohlhausen theory (reference 2, p. 109), fixing the velocity profile, and correcting the density through the boundary layer for the effects of compressibility. For the turbulent boundary layer, the momentum equation was integrated by a stepby step process in which a fixed velocity profile was used and the effect of density changes through the boundary layer on the ratio of the displacement thickness to the momentum thickness was ignored. The problem of commuting the momentum thickness over a body of revolution for incompressible flow has been treated by Young (reference 3), who computed the momentum thickness of the laminar boundary layer by a stepbystep CO7IVDE.ITIAL NACA ACr :*''. L6A22 con:,'it ; :in in ',I Iih an e:tens":on of the Po l, usen r.ethod wa. used. 'I'e thi. whch a fi:e'J elc.' .it ; fil. in ths ior:rentur. equat ,ion f3;i = t,..',j, cf' :e'..' l: ti.:.. ,:'er. ".1 ,. for a :d. off eluticn smrz io. In order:. to sj ~tsntiate the 9.. 2:r Ci L th .t F in friction fo.ru:.ls a ,s r tr rl..i ent fl]o:. aloni flat pr tes rr, a he usE:. in t'r.e 'n ,,tf:'t ion f n m.:.nt ur' thicknesses for flo,_, wvith iresc:uLre rai ients, r.liferLnncs : to o are cite' In thjse r f i r nc'es, I.c.C S r re.Si: nrt ',t':,een calcu. latce an,1 ey:,:' rire:',sl rer' ts 1 'A,: :ener.s]1.r ,obtained Slt!ou.,lh faii;rly '" ajvFrs: .reC E s.r? c;9 j.ts were present in man1' 3f the cRes. The as s .. t :.n that thc .iorrentrPi' thic'e ncl ess n Z I be coir:'lt:. :ei to a c ].C' 9 : :,': 'i: iatOto, 07n f1.r:in1 the velo,'itr orofile 'o.i. inj; inter tiocn is su:stsnt iate t c:, the v,'rk in refer.rcnc::'s s sid i, and by rlT; t in rrCc [erenc,: 6, which h contains E. ce,,:maro._ bc. 1 'een the co:m 'itedc: arn e 'eri mental : .'; iec if t mrrm=nt'r thick' s over the entire chcrd. That the sliinfrict ior cc cf ric ien:t for tirb!ul.nt flow is inde.'endc.nt cf !'ach number is established b th;. 'Aork in referenc s 7 n F'rc. 'el i'eference 7 rre; ents e.:pe rime.nt al d t t f',or turbuil rnt "icv'. a Eies ,hi,. show th' t tne vel .city profiles for lisubs: 'n.LC C O.Fri Zsi I e flow and tie skinfri':c ion cor fficienti for nbs.:iJc and 'surer so.liiC Corr.' ressir:le flow '.i not dififr a cticeaaly from those for ineir. sle flo ThecJdoi en and Remier referencee '} b.; xer.lriment ifr .lth ro'.tttinC 'is .:,., a'iov;ed that the kinfr Ltion coeff icient for turl:ulent boundary layers is irZndeFp ndent of M!'~i.:h nurmier. ITeena.r. and Ireaman referencee ) after *:erforming e':.riments \ ith pipes, reached ccnicl1sion: that c id not cor.trailct those cf references ,7 lnd i. a cont nt i eqiati:'n relating Pc to \ and E b slo .e of' ;clocity I1st1i tirit on ed ddrao coefficient '.er unit span CCITTDTPF 71~IrTT C O'F' ID' TI AL ITn A AC1 NTo. L6A22 co velocity of sound in free stream C, specific heat at constant pressure, footpounds per poundmass per degree TF ratio of displacement thickness to momentum thIickness (/e ) K constant k constant in skinfriction formula L length of airfoil, body of revolution, or round channel; measured along chord or axis of revolution Mo freestream Mach number m exponent in formula for boundarylayer velocity distribution m' particular value of m n exponent in skinfriction formula p static pressure R gas constant RL Reynolds number (UoL/Do) Rx Reynolds number based on length of plate (TJoX/Uo) Re Reynolds number based on momentum thickness (u'0/J) r radial distance of point from axis of body of revolution or round channel rt radius of body of revolution or round channel rtax maximum radius of body of revolution or round channel T .absolute temperature To absolute temperature of free stream T6 absolute temperature at edge of boundary layer T T'DT.:T' AT, CONF IDENT IAL 7 ..IA AC:R :TI. L6A22 CTIFT~ITTAL 5 I velocity parallel to x. at cuter e ,e of boundary IU, frx' est re= am ve locity Uxy al of at station at ich value of is obta "ned T. v.lue of TU t x = D vj V va. lue of 1U chosen to make value of I, a may imumi \/ U1 v.lue of cT at a :0 u velocity inside boLundridr l,r .annd oarollel to s'. .r a c w ;e:*.l"nent inr fcrr."ul a for viscosity x distance :rure alng surface from forward _.stl rn t ier. cint x0 po:si ion on surface at be..:inin of intecgrtion y distance m'c ';r,'D norm.; al to .x a siol. e b. t'vree t' c'ent to s'.ll9re cf bodyJ of revolution or round channel ancd ax:is of re vo1 ut on F = rt + i ccs a ~.1 val..ue of a st x v ratio o" sE:,ific heat at constant Dires1ure to s; c i fic hea t at cons c!n t volume 6 nominal tVhiclkneos cf bojunidary lDyer r 5e' cc 1 1 t tIJ n Srmomentui thickness P" i1 u 91 value of 8 9t x, CC. TPP T IAL PACA ACR Ho. L6A22 J(U/o)2 p. coefficient of viscosity kjo freestream viscosity u cinematic viscosity at outer edge of boundary layer Vo cinematic viscosity in free stream p density Po freestream density p5 density at edge of boundary layer To surface shearing stress = + 2 (y 1)Mo2 S= rte cos a i1 value of 9 at x0 d" =J (1 1y dy : f (1 PU e Subscripts: c compressible flow i incompressible flow ANALYSIS Momentum equation for compressible flow about a two dimensriLcpi tboo . hr. rc'nd r.yi yr. ror oe ntmi equation for twodimensional compressible flow (reference 10, p.l.2) CONFIDENT rI A CONFIDENTIALITY NACA ACR To. L6A22 COC~i'IlT TIAL 7 s i v n, henr the sttic:,res ur'e 'iris t.ion across the bo.in' r yr is nre li i e, a s / v'' d, *T 7 l U . c (1' FrThom the cua3 t ,n 1r'",'t i for c "'r." e s l it',isc fio" the rt. 1 O ion *..Yct,r., en the ;el,,.,, t; 'II, r e.. .re for CIcr."e=: i T,? 9 S ,'. :, Theni b. ;1sc of i tIhc ;cr'ati.. te tionr forP and the Ifi.L t ion f'or the ci l.ci .. c it ti ': ness 0 0 :: j*, (I ') equa.tion !, 1 ca:n ta ':. rI.ttn i, t'e for! ci.:en in refer nce 1 ".The .riC .e: ufse.d in th, .i.: ati,'r .f equ t on (2 were the conservation o.. mass ad aJevtor's law of moti'n. Equatio. (1) s thereorc a ?uic.,ole t, both subsonic ,nd. surers.nic 1. T. H + uti not, owe. er to be i', e r e whe assntions c.C the bof undcriass a.d ;rer ti oy "a; not be CCFITi'TI 41L SHA CA ACR io. L6A22 ;: rientutr equation for comnressible flow about a body of revolution. The boundarylayer omrentum equation for comnressible flow about a body of revolution (reference 10, p. 13) can be written, when the staticpressure variation across the boundary layer is negligible, as pu2r dy U  j 6x Jo pur dy = Tort r dy O I r r = rt + y cos a where rt and cos a are dependent on x only equation ( ) ""7 be rewritten as Spu2rt dy U  \+ x JO 6xJo +^ J p u2fcsyay 'Lr J (fig. 1), purt dy pu (cos a) ydy = x JO rt dy C I cos ay dy rtTo If 9 and P are defined by the relations rt9p6U2 pu(U u)rt dy and C(cos a) pgU2 = o pu (U u) (cosa).y dy and the equation of motion for compressible, inviscid flow 6 = PIT6U 6x 6x CONFIDENTIAL ix J Since COl,. ID:,: TIAL ".'A _.'r: D. L6A22 CO iTfC 1lITIAL is ued, th._en tlihe morne1.nt.im equation bec,:mprs pUrt dy ' C L 0 purt d  (rtrspu2) pOTT ico; aO.' d / j 1 put beso) dy . "i &J   cs O.p,. I = rtro Lt J L; and '": .r; be dief re:i ; the r"el=< i:ons t rt :.'p t= r p1r' pi)rt d.n and ('e" C. .)p6TT S (  !0 !p;, u) (ec ) ,: c.'y so that the trl.eiL rr: :i':ati'n be, ''.es + 2 ccs 2 + i P P;F ici 1 ,rP 0 1 K Q+ r ^^ ^c + 2) mo= StT 0, (rte + ':os c) = T T In cr.''ier t .r i t: r " : .c th e .or entum equaticrn f: : ct .:.re si ie "C.:'r a'..i t t. cd :.f te.oir icr., equation ( '. hu.u a be '. ritter in the s. e f,rrn as that for. t'"'o. i,",'ersi?. .i.c ca:,r. s.:;: 1 1T ._l eou.ati:nr (i2)). The ar'i.,xr i.st,' tl'i i tter'fcre !:cce list .r fl'.: rn the body of r e.Toi. ., i,,,n trtr(F + 2) +c s C) + r Cn? 'TTrID.'TTIL  T T 7 Then 6'  (rt dx. ?i(ti (4) = (:, + 2)(rte + C cos a)> 10 CONFIDENTIAL ITACA ;A^ ITo. L6A22 or 2 cos a~ 2 1 + 2i) rte \iic +2 / = K 1 cos a 1 C rte In order to determine the magnitude of the ratio + 2 Sthe assumption is made that the velocity distri Hc +2' butions through the boundary layer may be approximated by power curves of the type u /y 1/m Vt'hen the flow is incompressible, the definitions of *, Q, and Hi may be used to obtain S_1 +m and 2 +m m Therefore i + 2 _i 3m + 1 Hi + 2 3m + 2 for incomnressible flow. From / F\y/y dy = f1 F 1 dy do 2 Jo it can be shown that M m=2m H Sm=m m=2m' CC' TDT17 "IAL I.'A .'?? 1o. L6A22 CCiFF D.' TIAL 11 flar c'.i:.rezs 'bl as ."ell as f or inc?~: i rIes i ible flow.  js., cf and ft:':re 2. vhi,h is .isc'.sed later in connection iTAd S+ 2 the ef t of c .neszziClit c ; e rale K, + 2 Caii be ev T..ate fr cn. e _sble f'.low. i r all CAse3 of + 2 uinsearaited f1cv.. th'e vle f the r.tin iffers H + 2 frcii unit. b less } n 1C .'c:c ., f 1or r iost cases, bI' S a c,,h3 , rou hl7 1.' 'ercnt. Thus, een .:en is not a ' fr, . .;i Is not_ small frPcti'in, t.? r3:rc:i.mwtica :h?.t 7 = 1 is not far fri trius. .1er, in tadd.'ition, it l noted that t tte a .ro i and. i Ltler.e ',,re s3all over mcst of he boed:, a Fe apro i:i mraticn th.t = 1 is scorv'i:' le. If K = 1 is used in eq Lu tion (1), t :e r.ies Ir~. + n cos c + + 2 rt + Pos a + c to + cos = _x_ "_ x\ +" S,_Ox rtTo or u titu tlicn of . r' + ccs a ; A. + 2 0 1 c rtT,__, + +  5) o.:: U %:" pg o , p,,_U This *'.atio'f for' irco:nr.ess icle .f'iLo.: has been given in re :e rene .., r, e e n, 2 Ter principles .u.s. in ncLe erivatior. of equation (5) ;vere the conservation of mass and .:ewton's law of motion, The apDroxim.ation was made that. for flo.: on the body K = 1. 20 IIDE ZlTLAL NACA ACR jNo. L6A22 momentumm equation for compressible flow over inner surface of a round channel. 7fhen the flow in a round channel is such that the flow throughout the boundary layer is approximately parallel to the wall and a region exists outside the boundary layer in which viscosity has no effect, the boundarylayer momentum equation for this flow is the same as equation (3). In the momentum equa tion for flow over the inner surface of a round channel, the pressure change across the boundary layer is assumed to be negligible. All of the symbols in the momentum equation for compressible flow over the inner surface of a round channel have their meanings unchanged except for the distance y, which is positive when measured inward from the wall so that r = rt y cos a The derivation of the momentum equation for flow over the inner surface of a round channel follows the same pro cedure and involves the same assumptions as the derivation of equation (5). The derived equation is _] + 2 6U 1 6p_ C rtTo 6x U 6x Pg bx p6 where S= rt O cos a Effect of compressibility on Hc. Since the ratio of the displacement thickness to the momentum thickness H, occurs in the momentum equation, the effect of com pressibility upon this ratio should be considered. The equation for Hc is c CONFIDENTIAL CONFIDENTIAL IACA ... 0. L6A22 T.ID'TT IAL 1 S i t ari aton u the .r.unriar laern S.r t l.. 'rm the ve loci;ty i t'"rition th rou~.h the ':ou'a"' I'J er by r;str: acting the treatment to the case in i.lich ".ere is Irn h? at flo t'Luch th su'face and the ef. ctive Fran.ict nur.be s eqv.l uc nity. The eq .iat ion c i' + ccL*strt t is ue r 4. iea.li to fl .,.' in he :..cui 'ur'y, la'rer. ,,r,_e:"L t _e i ...itQ,"..1 res i:i;vt .c:n *C. .at the stat c n.iesz'.a ,.'a"r_.:t r. :.n ' ; r ;h the bo, xn:ia:r' l_:er.. iz ne*li.:ihle, the I j 9. e S p p rc li ' 0 f' chrcn induced tL '1 , Ii ia 1 <  :c: r 2 ").T ' ,I5 1 i + Y J3 i o  0 b " i 1 3. L \ 3 I ,.  L: T.AI TACA ACR No. L6A22 1 + X! Mo2 , r 1 2 1 + 2 bM *141 (TJ\2~" s where 2  =1 +o Yr 1) 2o2 then 2 By application of 6/65, then of the same procedure to the definition 2 U, (8) Large values of \2 mean s,iall values of Mo. The assur.tion was made that u /y\l/m and values were chosen for X2; 6'/5 and 8/6 were then calculated for a range of X between 1.5 and 14.0 with m 5, 54, 5, and 7. The curves of 1/Ec are given in figure 2. Since 2 +m fi  m CCOP i '"'TT T L Let (7) CONFIDENT IAL IAC A .Cf No. L6A22 CC,'PrIDrT; TI Ul. 15 an:i. ir.ce I :pe ? r in tl'e mn':; nt.rI e.qusatioin, the curves of figure 2 sre designTiste. by,r valus of l rather than rn. Pc,. er c ,rv.s sre usec' *rcrily for c. orvenl en1 ce6 n corriA'.ting th : e ff c.t .f c:, :. :r i b3 i] t H. The i.i of Fc for the Pi~ius f'iet:late .irof'il for l msn; r b o. '. ,'sy 1 :. rs i f rnce 2, .*' b een cor .ute.; for 'sriou. '. i ues of \ and s Prandrtl n.irrumber of iniit; t :v u , of equ atiins ( 7' d (ii). Ti'he cornm utations wer re. *.t tii th e. "el the r .el ..t" :i.St iliticn for lamin:sar flov o''er S fla.t o ~ 3.:t = (refern 11). 'he HCe ''.'a ;:. tte, sa air nst \, the results of both c,. minput ai tions w'r ;r s r ctical. 1 icer.tic]. ftr C il .i eq s of . On'1 th resu. jt? :bti;,n.: fo'r the 31 2rsius profile are there ore pr.rsente : ;i fiure r. Eq'. ti onl ( c.i 'F sov that 1 tlhozh the .elciit distributtion t.iroi the be wilEry1 ler 1 s med to be indeend.ent of s.: citln .lc nrfc th s.rfce, He may var with sui fs, .c sit:.:' .:eO'.s.? cf it ' : ': .nder e on . For intera tior. o.f the rmr. cnturr uj ti n, the de :enridence of B .n \ r',r be tsL.er, into acou'. bv a .ro:i.r t ingr the curi.u"s o f f'fi.rres 2 sri j 7 i.i. tio.n c + H S 1 1+ = ,. ] + H i (o' T I h The vl'es s: f a chob.sen to fit t,e c rv.es of fisc urcs 2 and 5 .'ith s efficient accu'jrs;y c ver r,,ost of te srane of S ar Ie lGotte; a inst Hi In fiLire '+. Inte.rato.:: f mor ntu e .. ic n f r t .,d nen sional flcw. P beforee e.'u:ofti.2n (2) .'ar ur i:.terst 'd, (He+ 2), 1 'o Tr i S, a,' oulj e r' e : fi.r.ctio s of the Pa ox p 2C number, sn the momentum h. Th term He + 2 is replaceFd by its ec iv'lenr . + (I + 2_ Br se 0 . C 'I I '! T P` T TM NACA ACR No. L6,22 of the gas law PF P RTg the equation of motion 6p \u 6x P 6x the Bernoulli equation for compressible flow TT2 Uo2 cPT6 + = coTo + and the equation for the velocity of sound Co2 = (y 1) cpTo the term 4 P can be written as Pn 6x 1 P6P _ P6 6x Uo I  T o To The local skinfriction coefficient P6U2 as a power function of the local Reynolds n on the boundarylayer momentum thickness by k k  P5o k Rn psU2 Rn9 U (10) is expressed umber based (11) By analogy with the work of reference 11, the viscosity and density used to calculate R9 are those at the outer edge of the boundary layer. The viscosity at the outer edge of the boundary.layer is assumed to be given by Ro \To/ P = /61w 110 TO (12) where w = 0.768 for air (reference 11). CO0tI T..nTIAL COI.TT FI DEIT T iAL v v HACA C. '.. L6_22 CONFIDENTIAL 17 The densit.: t te outer e'oe of the boundary layer is ePzJ ?I \i ri (l1, P6 3' 1 in :'iLci the flc. ov .t'ide the boun; dalr y la'.r is ass'.u.ed to be s i b.ti,?. '.itions (12; pid ( .1 ) are used to. give tr ere ir v s costt. in c.:ustorn ( i1 as funct icn :'r the velocity tiL tio '101. ver tie bc sn nd the freestresai Iv .' r,"Qr, r. Z.ouatiur (2 I t';he then be vwritteFn r s , ,. / \  9 a \u :,'' U ,L .. :, k 1:,n' a (, 1; / I  6x 'I _v A) ,i r L F oIl T j This equtlicn is d ifferef't~l equation of the Bernourlli type. i" 2n it is m*cde line 'r n n1d inte 2ra te : ': staE ndard methccs, t: e result is F ' \___,'L. P _L _ .L (f + ) L (/ ) 1/ n' I "i  i 1+I 1 \ TJ  J I Th i s /c I s 1 P+n 1" :a 1 k i 1 \L/ n \c. C ix '*1' ( I+n+j + l+ 17" 1 n CO'TID:FTT IAL ITIAC A ACR HOi. LoA22 where 1,/L and UI/Uo are the values of e/L and U/Uo at = . The value of is always greater than 1. L L By use of the Bernoulli equation for compressible flow, (o can be shown to be always less than hen Mo approaches zero, equation (14) becomes equation (1) of reference 6, that is, the integrated momentum equation for incoumpressible flow. Trt. ,r 9.t i. of 't.,..rntr rr, r f':r flo '1. over a body ,if 1.';,'. C' ho. .' n. ",, a_ ; . : *. .ii s Cj :lCc _U"Fi Te are the same as those for the twodimensional case. The equations for c, .T, and 6p are also the same o P6 P6 ex as for the twodimensional case. The expression for the skin friction, however, involves the apprcximation that 6 can be replaced by . This approximation is rt equivalent to neglecting the term cos a in the equation 0 Cos a rt rt Q 5 Since the value of t cos a is always less than 9e, it follows that the approximation is justified only in 6 cases in which  is a small fraction of unity. In rt regions near the tail, therefore, the accuracy of the approximation for the skin friction may be expected to decrease. The momentum equation for flow over a body of revolution (equation (5)) indicates, however, that the contribution of the skin friction to the boundarylayer thickness becomes less important as the tail is approached. The approximation that 0 can be replaced by is rtTrt therefore allowable and the term t0 in equation (5) P6U may be written n+1 n rtTo kt n p(15) pgTU2 pnun C r T17 T'r 7 mT T.,L CO~T R'IENTIAL FACA ACR io. L6 .'1 1 aps rtTo VWhen the terms Hc, P 6x and are replaced by PO 6x a PgU2 equations (9), (10), and (15), then equation (5) becomes TUU  ~U 2 D o c+ a t. Hi + 2 To : = Sx Ft ',U r 2 UG \t Uo j *^ ^ rtl nkn This equation is a di ffe.ern:t ia e iueitn of the Bernoulli type. '."hen it is macne linear srnr inttz:rated br standard method s, the result is i \ fa i ,\ , ~ , . . . L 2' 1 xL I J k(1+n) / L \l+n R k)n tmax \ Y1 rt '1+n. +i)(i+n +'+ilr ] , l+n) T /I i Uo 'i (1+ri) J, i71t1raz L wh re  ard 2 tma x U/Uc .t  L L i+21 1 2, 1+n 1 V 0 1 Sa( i+ni+n \, 'l IT/,/o re the values of 2 and t max CC';r, I'D 7T TA.J CON IDENTICAL .i 'T'1L 'j)'" NACA ACR Ho. L6A22 Integration of momentum equation for flow over ifrer surface of a round chennel. The equation resulting from the integrationn of equation (6), when the same procedure and assumptions made to integrate equation (5) are used, may be obtained from equation (16) by replacing p by /. Tn the equation for 1/, the quantities denoting reference conditions, Uo, rtmax, and M,, are the condi tions at a convenient reference section of the channel and the length L is any convenient length. The definite integrals occurring in equations (14) and (16) and the equation for i may be evaluated by either an analytical or a rraohical method, whichever is more convenient. DISCUSSION Before equation (14), equation (16), or the integrated equation containing can be used, it is necessary to know the velocity distribution over the surface, the con stants in the skinfriction formula, and the value to choose for Hi. The velocity distribution rust be that for the Miach number and Reynolds number for which the com putation is being made. Skinfriction formuil.s. For the turbulent boundary layer, a power function of. Rg is used for the skin friction formula. Because references 7 and 8 indicate no noticeable effect of MKach number on skin friction, a skin friction formula for low speeds may be chosen and approxim ated by TO k psU2 R n as outlined in reference 6. One formula of the required form is that of Falkner (reference 12) To 0.006535 p5JU2 Re1/6 If the skinfriction data available are given in the form of cd ~ Rx, they may be converted to Re by use pgU CO!T. IDE TIA I, iAC. kCR 0 0o. L6A22 CONFIDENTIAL 21 c.f che rel"ti "u n  Tr 1ic d \ r alr:LnE r bcndar. v_ la'yjr. ." thc'u h e. .,tin.ri ('L , equating ic. ,nd: the tqu .ion for h'.e been .derived. for turbulent ';1oun'ci'ry L''ers, these eqiati cns mey b used for the aprox.T'i1e c',O s ti .r1 of ar in lanil nar tcUndr ': ly er . t .hS, Frr ri.A e choice of a value for Hi an. ,..f a sini rict' on formula. Althcurh by use of the sl:infr ict.rion relaticn f.r the S i&..isi flat plate profile the sffect cf rE. sure .adient o.n the skin fricticn coefficien is nrlectrd, the rror introd uced is small so. lon s Is the .verae oressure gra,'ient over the extent af tha iaminr bo.; dsr, larver is Erriall. Thle skin friction ieiat ion trt n is TO 0.220 p T T2 R, 6& where 1: = 0'.2 ad n = 1. Hlthclu. h sma T! l decrease occisrs in k 1 s Ltie a3h ni,ber incre s i refercnce 11), the value cf 1 fcr inionr'. ssitle fl.,' (0.22C') may be used in ' of t e a, :.r,'c. i cTio l irs .r ,naile coic rntr,, the sk .n fri:ti:n. Choice of ji. TIn refrer es ", and )., the .a'ie of t he >cr ndary larer shI a s.nr ter was r estric 'te to 1.4. If eciuati:n (1J4 is uJst, is i'en a 91 arbitrar.. increaent Hi, and 0 tren the chance in r.a'Y be ei'Ten by kr. \L Y T_ , where T. !ies bc.tveen the :rr'xim~r and rrinlmt.um velocity in the interal ::, :x. The term i] has been Cc;F rID TnTII.AL N.ACA ACR :D. Lo422 eval.:uated for incompressible flow and a straightline velocltg distribution U = Tx0 bx The expression for becomes Si Al"[,(Hi+l)(n+l)+2+ i (n+1l) (!n+2 U II '1 It j ^ri~i ~ tn+l)( + + l 1 ii I (i + 1)(n + 1) + 2 (17) (i + 1)(n + 1) + 2 + AHj(n + 1) Although equation 17) is for ircompressible flow and the exoression for  x is only a: roximstely correct for cases in wic.  0, equation (17) indicates that TE excessive errors in 8/L are possible if a in equation (9) is made to equal zero and Hi is replaced by Fc andrestricted to a constant cvsseed value when the actual variation of Hc is large. Irnrea.sin the body thickness, the lift coefficient, or the M~rach number increases the variation of Hc over the :.iface. In the integrated equations the value of He may vary over the surface but that of H is assumed to be constant. Because Hi can usually be estimated to withinn 0.2 for cases in ';Sich flow separation does not occur, the error in /. that ?may be caused by a poor choice of Hi is unlikel' to be more than 10 percent. "hen the velocity increases in the direction of flow, the value of Ei may be taken ss 1.2. When the velocity decrcses in the direction of flow, the value of Hi should b increased fro.n l., for cases in which thie total change of velocity is aboii 20 prcent of the initial velocity, to values near 1,71 for cases in which the total change in velocity exceeds 0 percent of the initial velocity. Because the velocity profile for the laminar COC "IDEHTIAL CONFIDENTIAL N.CA A.Cx No. L6A22 CrnFIrT:ITIAL, 23 bourndi .ry 2. ~l;r has been res tr. te.d tc th"e ilasius flat plate profile, Fi is equal to 2. Fo,2 the larri.nai b..iu.ry lawyer, a then has Yte value i.. Full thickl.ness3 of brirndary la"r. Thse 'ui 1 thickne s of r.e t...rbulA r.t boE'nd r: ." l1ay r in tch :ase of" t'.,o dier.sioral fle''r mr; te obtainec y. .'se of the relation For th.e hcd 4 f re'rc.l1t.ic r the, f'.ill thickness of the turbu.lenu L o.:id r.; _ay r ny a tsine., after rtmax is comT;_:,lted, r.. i.use cf c:Fe rel stion 1 + I + 6 2 cLs C trm 2j ) cos a. \ C ((18) Curves of _/6s and 62 aSginst K are given n figures 5 and 6 for four vsli .s of H . The va:iattin of 9/6 for the PBslsis flatnlate profil is ,i r.n in fi,'ura '7 for use i ccm:u:ting the full thickness of the lavina r bo:L"..r layer. For flow abcut body of rev.oljticn, C/62 ay or n. lec r., since the lam inar bo'unler:/ l19er's are usJally thin. The expression for th full tLhick: ss of the la iTnar ::und.sr:y layer on a booy of revolution thoreifoe becomes P f r \( rtj. \2 t cf Per fi':. ,.:ve r the inner sr._:face f s rond channel when the .b, und.?:, 1 7 v is thin, eauations (1K) and (1) are rer?.c e by v ()  x ) cos a rat tm ax tCmax (2. r2. : cos a(20) C'T, IDENIT ,AL . C'TID':TTAL NA1CA ACR No. L6o' and rt6 = (/ Nj (21) r max tmax ( \ 21) In order to obtain a general statement regarding the effect of density changes on the momentum thickness, com outations of 8 were made for a linear velocity distri bution and a range of Mach number. The velocity distri* bution for 0 x ( 1 was defined by r U x S I1 + b Uo L The only variable was Mo. The curves of 9 against Mo are given in figure 8 for b = 0 and 0.2. The results indicate that the effect of density variation becomes important only at Mach numbers exceeding unity. CONCLTTSIONS A comparatively rapid method is presented for the computation of boundarylayer momentum thicknesses for flow over twodimensional bodies, over bodies of revolu tion at zero angle of attack, and over the inner surface of round channels all in compressible flow. The computations indicate that the effect of density changes on the momentum thickness in flows with pressure gradients is small for subsonic flows. Langley :'o;.:~iorial Aeronautical Lacoratory National Advisory Committee lor aeronautics Langley Field, Va. C c;F TDTUITIAL IHTACA ACR No. L6A22 CCIFTriiITIAL 25 1. Yo'.ingp, D., and '"interbc ttomn, I. E.: Iote on the Effect of C.ev;:re ssibilit' on the Profile Dras of .ero..f'oill in Lhe Aibsence cf Shock "Ia'.'es. Re'. 'o. A. 5) R .A. ( Iiti.h) :.ay 1 110. 2. Pr:ndl, I.: The !'vechpics c.f iScous Fluids. 'cl. ITT of kerc.dJnr mic' T'ner' di'. '~, sees. 1!. ;.nd 17. .'.. '. D'ur.?nd, 6,<., Jl] i'i.? S'.r,,/?F., ( Perl.in), 195" , pp. 8~ :'lUIR anSd I C21 2. Yun D. : LTh Calc:.ll:in of the Tc.tal and Skin F'rition DragEs of Roies of Revolut'Ion at Zero Iinci'i. ,e. R7. .:. 3 '7i I'iti lh A. .C. 1959. I. Squire, P. :... n Y .n .2. ..; The C sT l c..ia tion of their of Aerrfoils. '. f i.A 1i., L.r tish t.. ". ., 1 h'. 5. von Doenhoff, Al'ert ., a<. Tetrvin, aeal: D nation of G a?'r ral el .aionrIs for tr: .F Fe;ii, ior c,f 'P ut ul rent E:;t.d'ry L ay7 '. AI CT ( :1 1 '31 19.. 6. T'etervr.n, :Tes ]: A "eth.od fior the Ranid Estir.iation of Turlile .t So...ndarrLai er Thicl.es .es for Calci.lating Prc.file Drr9;.. :.: ',. A,' 11 L lJC14, 19 4 . 7. Fr'se]., .: ? cc'. in Smooth Strai _ght Pine. at Velccities s':ove ar cnd blow 3Sonc '";elocity. ITACA Ti ,u. 2 l S.. c. The 'orern, Thecdore, aid Re ier, K'thi.r. Experiments orn Dr a. cf Rvolvir. Disks Cylner.s, 9nd Stream l.ie Reds It Hih Sceeds. ?IC, \CP l. [14_F 19,? . 9. Ieernar, Joseph F., ar .i e'.ran', 9 Fr: *t P.: Friction in Pi :es at ES.,,;:ler .i.. .c n:r .Si bts v 'tic Velocities. liACA TV ,. 9&.5, '?' 10. Fluic' titicr. Panel of the .oror,..tiCa] Research C ir t e a:;:.d= d Cthers: n "ern Die\lor.en ts in Fluid Dr.na'n i Vo, T., S. iold:rten, ed., The Clarendon Tress (Cxf, rd,, r d'. Tr. . I LrTT ' L i L AT C C::7I)TIAITT liNOA ACR No. L6A22 11. Brainerd, J. G., and rmnons, H. 7.: Effect of Vari able Viscosity on Boundsry Layers, wi t. a i cussion of Drag Measurements. Jour. Appl. Mech., vol. 9, no. 1, March 1942, pp. Ai A6. 12. Falkner, V. M.: A New Law for Calculating Drag. The Resistance of a Smooth Flat Plate with Turbulent Boundary Layer. Aircraft Fnlr.eering, vol. XV, no. 169, March 1945, pp. 6569. CC' IT'IT IAL NACA ACR No. L6A22 Fig. 1 us 0' OU 4 Z. z 8 1 0 0 0 o. Z UJ w W o0 0 u zo Y4 0 *H r74 Fig. 2 NACA ACR No. L6A22 %a w . II M. 0 S" z 4 .   = a oo o O z e    4 z  t \10 \ \ 0 , 0 ^ < r ___h__ V4 NACA ACR No. L6A22 Fig. 3 0 41 43 Cd 1I 41 C 1'1 H 43 ca PQ cd 0 14 11 :4  4o d 0 N4 NACA ACR No. L6A22 '00 si 0 p 14 d 0 q4 Sdt Cn SO 1k N) u N' Fig. 4 NACA ACR No. L6A22  II 0 A a ZZZiIiZii Iz0 iii zn' _ __ _ U 9 I. 9 K 8 z L. o   z 0 Pl I I I I I I I I ~ N ~ N N N N. Fig. 5 .1I Ir) Iri w 43 o 0 o 4J ra 4 43 0 p, to O I*4 NACA ACR No. L6A22 _  _ a. P" 0 U 2~  So 04 I II u:~~  I. 2     o L::! I.  0. cIq Fig. 6 g fr N *"I 4, CM I %H 0 o r4 rM4 I h0 Tl NACA ACR No. L6A22 U O Sal 04 zo Z O z   ZLL __.  U. r ,H CH U, 0 r4 0 (l 0 S a, 0n 0 64 F4 r F ) < bLa Fig. 7 NACA ACR No. L6A22 .o oze .oozo .0068 .0024 .0020 .00/6 .00[2 .0006 .0004 0 CONFIDENTIAL b 0.2 CONFIDENTIAL /o NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS Figure 8. Variation of button given by U = Uo constants in equation RL = 107; w = 0.768; = O. B/L with Mo for velocity distri 1 + bL; 0 < x/L <1. L Values of (14): k = 0.006555; n = 1/6; y = 1.4; Hi = 1.4; a = 0.78; Fig. 8 UNVERSTY OF FLORIDA DOCUMENTS DEPARTMENT 1 .RSTON SCIENCE UBRPARy F.O, BO,, 117011 GAINESVILLE, FL 326117011 USA 